TITLE: Poincare duality and the $A_\infty$ structure on cohomology
QUESTION [28 upvotes]: If $X$ is a topological space then the rational cohomology of $X$ carries a canonical $A_\infty$ structure (in fact $C_\infty$) with differential $m_1: H^\ast(X) \to H^{\ast+1}(X)$ vanishing and product $m_2: H^\ast(X)\otimes H^\ast(X) \to H^\ast(X)$ coinciding with the cup product.
If $X$ is a closed manifold then its cohomology algebra satisfies Poincare duality.  This is a condition that refers only to the $m_2$ part of the $A_\infty$ structure.  There are things more general than manifolds that satisfy rational Poincare duality, such as rational homology manifolds.  In general, a space that satisfies rational Poincare duality is called \emph{rational Poincare duality space}.  
Obviously, the statement that $X$ is a rational Poincare duality space places no additional restrictions on the $A_\infty$ structure beyond the condition on the product.  
Question 1  Can one construct from the higher multiplications obstructions for a rational Poincare duality space to be rationally equivalent to a rational homology manifold, or a topological or smooth manifold?
Here is a second and somewhat related question.  Poincare duality says that $H^\ast(X)$ is self-dual (with an appropriate degree shift). Thus the adjoints of the higher multiplication maps make the cohomology into an $A_\infty$ coalgebra (with appropriate adjustment of the grading). 
Question 2  How does this $A_\infty$ coalgebra structure interact with the $A_\infty$ algebra structure on the cohomology?

REPLY [16 votes]: It was shown in the IHES paper in the 70's "Infinitesimal computations in Topology" that over Q   homological Poincare duality is sufficient for a simply connected  Q homotopy type to be realized by a smooth  closed simply connected manifold in dimensions  5,6,7,9.10,11,13,... etc.
 In dim 8 ,12, 16, etc necessary and sufficient conditions
to realize are given. These are tantamount to the Thom Hirzebruch signature formula holding for some choice of fundamental class where the quadratic form over Q is a sum of positive and negative squares.
These statements imply the answer to question 1 is: There are no implications
on the higher infinity structures implied by a closed manifold representative
of a rational homotopy type.